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Theorem plymul0or 20151
Description: Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
Assertion
Ref Expression
plymul0or  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  <-> 
( F  =  0 p  \/  G  =  0 p ) ) )

Proof of Theorem plymul0or
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dgrcl 20105 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
2 dgrcl 20105 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3 nn0addcl 10211 . . . . . . 7  |-  ( ( (deg `  F )  e.  NN0  /\  (deg `  G )  e.  NN0 )  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
41, 2, 3syl2an 464 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
5 c0ex 9041 . . . . . . 7  |-  0  e.  _V
65fvconst2 5906 . . . . . 6  |-  ( ( (deg `  F )  +  (deg `  G )
)  e.  NN0  ->  ( ( NN0  X.  {
0 } ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 )
74, 6syl 16 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( NN0  X.  { 0 } ) `  ( (deg
`  F )  +  (deg `  G )
) )  =  0 )
8 fveq2 5687 . . . . . . . 8  |-  ( ( F  o F  x.  G )  =  0 p  ->  (coeff `  ( F  o F  x.  G
) )  =  (coeff `  0 p ) )
9 coe0 20127 . . . . . . . 8  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
108, 9syl6eq 2452 . . . . . . 7  |-  ( ( F  o F  x.  G )  =  0 p  ->  (coeff `  ( F  o F  x.  G
) )  =  ( NN0  X.  { 0 } ) )
1110fveq1d 5689 . . . . . 6  |-  ( ( F  o F  x.  G )  =  0 p  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( NN0  X.  { 0 } ) `  (
(deg `  F )  +  (deg `  G )
) ) )
1211eqeq1d 2412 . . . . 5  |-  ( ( F  o F  x.  G )  =  0 p  ->  ( (
(coeff `  ( F  o F  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( NN0  X.  { 0 } ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  0 ) )
137, 12syl5ibrcom 214 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  ->  ( (coeff `  ( F  o F  x.  G ) ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 ) )
14 eqid 2404 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2404 . . . . . . 7  |-  (coeff `  G )  =  (coeff `  G )
16 eqid 2404 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
17 eqid 2404 . . . . . . 7  |-  (deg `  G )  =  (deg
`  G )
1814, 15, 16, 17coemulhi 20125 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( (coeff `  F ) `  (deg `  F )
)  x.  ( (coeff `  G ) `  (deg `  G ) ) ) )
1918eqeq1d 2412 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  o F  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  x.  ( (coeff `  G
) `  (deg `  G
) ) )  =  0 ) )
2014coef3 20104 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
2120adantr 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  F
) : NN0 --> CC )
221adantr 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  NN0 )
2321, 22ffvelrnd 5830 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  F ) `  (deg `  F ) )  e.  CC )
2415coef3 20104 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2524adantl 453 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  G
) : NN0 --> CC )
262adantl 453 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
2725, 26ffvelrnd 5830 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  G ) `  (deg `  G ) )  e.  CC )
2823, 27mul0ord 9628 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
( (coeff `  F
) `  (deg `  F
) )  x.  (
(coeff `  G ) `  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
2919, 28bitrd 245 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  o F  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3013, 29sylibd 206 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  ->  ( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3116, 14dgreq0 20136 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3231adantr 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0 p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3317, 15dgreq0 20136 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3433adantl 453 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3532, 34orbi12d 691 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0 p  \/  G  =  0 p )  <->  ( (
(coeff `  F ) `  (deg `  F )
)  =  0  \/  ( (coeff `  G
) `  (deg `  G
) )  =  0 ) ) )
3630, 35sylibrd 226 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  ->  ( F  =  0 p  \/  G  =  0 p ) ) )
37 cnex 9027 . . . . . . 7  |-  CC  e.  _V
3837a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
39 plyf 20070 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
4039adantl 453 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
41 0cn 9040 . . . . . . 7  |-  0  e.  CC
4241a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  e.  CC )
43 mul02 9200 . . . . . . 7  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
4443adantl 453 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
4538, 40, 42, 42, 44caofid2 6294 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( CC  X.  { 0 } )  o F  x.  G )  =  ( CC  X.  { 0 } ) )
46 id 20 . . . . . . . 8  |-  ( F  =  0 p  ->  F  =  0 p
)
47 df-0p 19515 . . . . . . . 8  |-  0 p  =  ( CC  X.  { 0 } )
4846, 47syl6eq 2452 . . . . . . 7  |-  ( F  =  0 p  ->  F  =  ( CC  X.  { 0 } ) )
4948oveq1d 6055 . . . . . 6  |-  ( F  =  0 p  -> 
( F  o F  x.  G )  =  ( ( CC  X.  { 0 } )  o F  x.  G
) )
5049eqeq1d 2412 . . . . 5  |-  ( F  =  0 p  -> 
( ( F  o F  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( ( CC  X.  { 0 } )  o F  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
5145, 50syl5ibrcom 214 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0 p  -> 
( F  o F  x.  G )  =  ( CC  X.  {
0 } ) ) )
52 plyf 20070 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
5352adantr 452 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
54 mul01 9201 . . . . . . 7  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5554adantl 453 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5638, 53, 42, 42, 55caofid1 6293 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC  X.  { 0 } ) )
57 id 20 . . . . . . . 8  |-  ( G  =  0 p  ->  G  =  0 p
)
5857, 47syl6eq 2452 . . . . . . 7  |-  ( G  =  0 p  ->  G  =  ( CC  X.  { 0 } ) )
5958oveq2d 6056 . . . . . 6  |-  ( G  =  0 p  -> 
( F  o F  x.  G )  =  ( F  o F  x.  ( CC  X.  { 0 } ) ) )
6059eqeq1d 2412 . . . . 5  |-  ( G  =  0 p  -> 
( ( F  o F  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( F  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) ) )
6156, 60syl5ibrcom 214 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0 p  -> 
( F  o F  x.  G )  =  ( CC  X.  {
0 } ) ) )
6251, 61jaod 370 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0 p  \/  G  =  0 p )  ->  ( F  o F  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
6347eqeq2i 2414 . . 3  |-  ( ( F  o F  x.  G )  =  0 p  <->  ( F  o F  x.  G )  =  ( CC  X.  { 0 } ) )
6462, 63syl6ibr 219 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0 p  \/  G  =  0 p )  ->  ( F  o F  x.  G
)  =  0 p ) )
6536, 64impbid 184 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  o F  x.  G
)  =  0 p  <-> 
( F  =  0 p  \/  G  =  0 p ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774    X. cxp 4835   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   0cc0 8946    + caddc 8949    x. cmul 8951   NN0cn0 10177   0 pc0p 19514  Polycply 20056  coeffccoe 20058  degcdgr 20059
This theorem is referenced by:  plydiveu  20168  quotcan  20179  vieta1lem1  20180  vieta1lem2  20181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063
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